Green’s Dictionary of Slang

bit n.3

SE in slang uses

In phrases

put the bit on (v.) [SE bit, the mouthpiece of a horse’s bridle, used to restrain the animal]

(Aus.) to extort, to blackmail, to force someone to do something they would rather avoid.

[Aus]D. Niland Call Me When the Cross Turns Over (1958) 33: A sheila comes up to me [...] and puts the bit on me because she’s hard up.